Quantification of Numerical and Modeling Errors in Simulation of Fluid Flow through a Fixed ...

157 Pages·2015·3.7 MB·English

Quantification of Numerical and Modeling Errors in
Simulation of Fluid Flow through a Fixed Particle Bed

A thesis submitted to the
Graduate School
of the University of Cincinnati
in partial fulfillment of the
requirements for the degree of

Master of Science

In the Department of Mechanical and Materials Engineering
of the College of Engineering and Applied Science
by

Annette Volk

B.S., University of Evansville, 2012

December, 2015

Committee Chair: Urmila Ghia, Ph.D.
Abstract
Detailed description of flow through stationary particle beds is crucial for the design and
implementation of municipal water filtration, material extraction systems for nuclear waste and
industrial water purification systems. Knowledge of fluid-particle interactions and fluid flow properties
through the bed is essential to design, but difficult to determine from experimental investigations.
Combined granular-fluid simulation methods such as coupled Computational Fluid Dynamics and
Discrete Element Method (CFD-DEM) have been used to bridge this gap in fundamental knowledge.
Able to capture details of the small-scale and large-scale interactions that are difficult to study in
physical beds, simulation findings have added great understanding to this field. Unfortunately, the
reported results are occasionally flawed by a lack of understanding, specifically regarding the magnitude
of numerical and modeling errors. Uniform reporting of error, investigations of simulation trend, and
proof of mesh-independence have not been performed for granular-fluid simulations.
A standard method of open-source granular-fluid flow simulation known as CFDEM is applied to
the simulation of flow through a fixed bed. The Ergun equation is a validated empirical expression used
to predict the drag force in fixed bed flow and this prediction is compared directly to simulation results.
A grid-refinement procedure, standard for publication of CFD simulation results, is applied to
the CFD-DEM simulations. The solution trend over the refinement range is investigated using the
frequency of convergence, convergence types, and the proposed ‘offset’ method; a comparison of the
expected numerical error and actual extrapolated solution error.
An optimal grid size resulting in the least amount of error is investigated by solution trend, drag
profile comparison, and the grid-refinement study results. Error is seen to increase in the simulations at
both large cell sizes and as the cell size approaches one particle diameter. A new grid-refinement study
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application that does not require analytical solution data is shown to be a good predictor of relative
error in the grid solutions.
Three drag correlations are applied to model fluid flow through a fixed bed. The Gidaspow drag
correlation is an exact representation of the Ergun equation, and shows high accuracy. The Di Felice
correlations is a continuous-function representation of expected drag force. The Koch-Hill correlation
was developed from LB simulations for fluidization conditions, and is chosen as an example of a poor
correlation choice. The grid-refinement study results are able to distinguish the poor performance of
the Koch-Hill correlation from the highly accurate Gidaspow and Di-Felice correlations.
The standard grid-refinement study is shown to be applicable to granular-fluid flows, and to
produce results that are useful for common modeling choices. Relatively low convergence frequency of
the grid-refinement studies is expected to hinder future application by requiring additional grid
solutions. This procedure is recommended for all granular-flow simulations since it provides useful
information which can prevent common modeling errors that have hampered fluidization research.

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Acknowledgements
First I would like to thank my very dedicated advisor Dr. Urmila Ghia for her tireless efforts on
this work. Her suggestions and insights have been crucial at every step. Dr. Urmila Ghia and Dr. Karman
‘Kirti’ Ghia have been vital to my development as a researcher.
Likewise, Dr. Chris Stoltz’s dedication to this work and constant enthusiasm have been vital to its
success. I am also grateful to Dr. John Hecht for his ability to connect every goal to the big picture to
keep my research on track. I would like to thank Jason Stamper for his contributions to the
development and practical application of this work.
I would also like to thank my U.C. Simulation Center colleagues and mentors, especially Dr.
Aravind Kishore, Sushrutha Reddy Gujjula, Santosh Konangi, Sandeep Madireddy, Sadegh Riasi, Dr.
Bernie Rudd, Fred Murrell, and Dr. Kelly Anderson.
I am grateful to the National Science Foundation, for its Graduate Research Fellowship
sponsoring this work under Grant Number 1102690.
I give my heartfelt thanks to the many people in my life who have supported me through this
journey. To my mother, Diane Volk, for her constant support of all that I do, and for continued correct
use of the term ‘hydrophobic’. To my family and friends, especially my father, Kevin, my brother, Phil,
and my road trip companion, Janel Jett. And of course to my scuba family including Steve, Craig, Ed,
Lisa, Po, Josh, Carole, Tom, Griselda, Mike, Debbie, Jen and Ernie!

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Table of Contents
Abstract ......................................................................................................................................................... ii
Acknowledgements ....................................................................................................................................... v
List of Tables ................................................................................................................................................ ix
List of Figures ................................................................................................................................................ x
Nomenclature ............................................................................................................................................. xiv
Chapter 1 Introduction .............................................................................................................................. 1
1.1. Motivation ..................................................................................................................................... 3
1.2. Overview of Previous Research .................................................................................................... 4
1.3. Objectives of this Study ................................................................................................................ 6
Chapter 2 Analytical Method ..................................................................................................................... 7
2.1. Pressure Drop Relations for Flow through a Fixed Bed ................................................................ 8
2.1.1. Ergun Equation .................................................................................................................... 10
2.2. Minimum Fluidization Velocity ................................................................................................... 14
2.3. Grid-Refinement Study ............................................................................................................... 17
2.3.1. Development of the Grid-Refinement Study ...................................................................... 17
2.3.2. Current Grid-Refinement Study Procedure ........................................................................ 19
2.3.3. Convergence Categories ..................................................................................................... 22
Chapter 3 Numerical Method .................................................................................................................. 25
3.1. Multi-Scale Modeling of Fluidized Beds ...................................................................................... 27
3.2. Open-Source Software Advantage .............................................................................................. 31
3.3. Computational Fluid Dynamics (CFD) ......................................................................................... 33
3.3.1. Conservation Equations ...................................................................................................... 33
3.3.2. Implementation .................................................................................................................. 34
3.3.3. Importance of Grid Size....................................................................................................... 34
3.4. Discrete Element Method (DEM) ................................................................................................ 38
3.4.1. Hard Sphere vs. Soft Sphere Model .................................................................................... 38
3.4.2. Soft-Sphere Model Equations ............................................................................................. 40
3.4.3. Time Step Value .................................................................................................................. 43
3.4.4. Spring Constant Value ......................................................................................................... 45
3.5. CFDEM Coupling .......................................................................................................................... 47
3.5.1. Coupling Types .................................................................................................................... 47
3.5.2. Implementation .................................................................................................................. 48
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3.5.3. Coupling Procedure ............................................................................................................. 48
3.5.4. Porosity Calculation ............................................................................................................ 50
3.5.5. Drag-Closure Equations ....................................................................................................... 52
3.5.5.1. Analytical Comparison ................................................................................................ 55
Chapter 4 Simulation Method ................................................................................................................. 60
4.1. Geometry and Boundary Conditions .......................................................................................... 61
4.2. Computational Grid Design ......................................................................................................... 63
4.3. Initialization ................................................................................................................................. 68
4.4. Velocity Profile ............................................................................................................................ 71
4.5. Fluid and Particle Properties ....................................................................................................... 78
4.6. Time-Step Values and Coupling Interval ..................................................................................... 84
4.7. Data Output for Post-Processing ................................................................................................ 86
Chapter 5 Results and Discussion ............................................................................................................ 87
5.1. Solution Trend ............................................................................................................................. 89
5.1.1. Curvature ............................................................................................................................ 93
5.2. Grid-Refinement Study ............................................................................................................... 97
5.2.1. Grid Combinations .............................................................................................................. 99
5.2.2. Frequency of Convergence ............................................................................................... 100
5.2.3. Offset Method ................................................................................................................... 102
5.3. Optimal Cell Size ....................................................................................................................... 104
5.3.1. Solution Trend ................................................................................................................... 104
5.3.2. Regression Coefficients ..................................................................................................... 104
5.3.3. Offset Method ................................................................................................................... 105
5.3.4. Alternate Methods of Optimal Cell Size Determination ................................................... 107
5.3.5. Choice of Cell Size ............................................................................................................. 109
5.4. Comparison of Drag Correlations.............................................................................................. 111
5.4.1. Solution Trend ................................................................................................................... 111
5.4.2. Regression Coefficients ..................................................................................................... 114
5.4.3. Offset Method ................................................................................................................... 117
5.4.4. Convergence Frequency .................................................................................................... 119
5.4.5. Average GCI Value ............................................................................................................. 122
5.4.6. Overall Comparison ........................................................................................................... 123
Chapter 6 Conclusion ............................................................................................................................. 125
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6.1. Future Work .............................................................................................................................. 127
Appendix A ................................................................................................................................................ 129
Appendix B ................................................................................................................................................ 131
References ................................................................................................................................................ 132

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List of Tables

Table 3.1: Hertz Contact Model Coefficient Equations ............................................................................... 42
Table 3.2: Equivalent Particle Property Expressions ................................................................................... 42

Table 4.1: Fluid Properties (Ambient Air) ................................................................................................... 78
Table 4.2: Particle Properties (Airsoft Pellets) ............................................................................................ 79

Table 5.1: Complete Results from Single Group Grid-Refinement Study ................................................... 98
Table 5.2: Averaged Results From Single Group Grid-Refinement Study ................................................... 98
Table 5.3: Grid Size Ratios Resulting in Least Error with Ergun Equation Drag Force Values ................... 104
Table 5.4: Grid Size Ratios Resulting in Least Error with Ergun Coefficient of Velocity ........................... 105
Table 5.5: Grid Size Ratios Resulting in Least Error with Ergun Coefficient of Velocity Squared ............. 105
Table 5.6: Grid Solutions with Least Average Offset ................................................................................ 106
Table 5.7: Average Offset for Each Drag Correlation Implementation ..................................................... 119

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